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Oscillations of first order linear retarded differential equations. (English) Zbl 0731.34080
The author considers the first order retarded differential equation \[ (E)\quad x'(t)+\sum^{n}_{k=1}p_ k(t)x(t-\tau_ k(t))=0, \] where \(p_ k\) and \(\tau_ k\) \((k=1,2,...,n)\) are non-negative continuous functions on an interval \([t_ 0,\infty)\), and \(\lim_{t\to \infty}(t- \tau_ k(t))=\infty\) \((k=1,2,...,n)\). Main results of the paper establish sufficient conditions for all solutions of (E) to be oscillatory and conditions, under which (E) has at least one positive solution x with \(\lim_{t\to \infty}x(t)=0\) and such that \(x(t)\leq \exp \{-\lambda \int^{t}_{t_ 0}[\sum^{n}_{j=1}p_ j(s)]ds]\) for all large t(\(\lambda\) is a positive number).
In the paper we find several relations with earlier known results.

MSC:
34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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